{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:42:44Z","timestamp":1760186564069,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,2]],"date-time":"2019-01-02T00:00:00Z","timestamp":1546387200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002403","name":"Taibah University","doi-asserted-by":"publisher","award":["2019"],"award-info":[{"award-number":["2019"]}],"id":[{"id":"10.13039\/501100002403","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Edge Even Graceful Labelingwas first defined byElsonbaty and Daoud in 2017. An edge even graceful labeling of a simple graph    G    with    p    vertices and    q    edges is a bijection    f    from the edges of the graph to the set     { 2 , 4 , \u2026 , 2 q }     such that, when each vertex is assigned the sum of all edges incident to it     mod 2 r     where     r = max { p , q }    , the resulting vertex labels are distinct. In this paper we proved necessary and sufficient conditions for the polar grid graph to be edge even graceful graph.<\/jats:p>","DOI":"10.3390\/sym11010038","type":"journal-article","created":{"date-parts":[[2019,1,3]],"date-time":"2019-01-03T03:36:30Z","timestamp":1546486590000},"page":"38","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Edge Even Graceful Labeling of Polar Grid Graphs"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3809-2521","authenticated-orcid":false,"given":"Salama Nagy","family":"Daoud","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah 41411, Saudi Arabia"},{"name":"Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El Kom 32511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,2]]},"reference":[{"key":"ref_1","unstructured":"Acharya, B.D., Arumugam, S., and Rosa, A. 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